If the washer won't spin the motor coupling might have failed. The motor coupling connects the motor to the washer transmission. It is designed to fail if the washer is overloaded in order to protect both the motor and transmission. Another common problem is the lid switch.
Motor Control Board
If the washer won't spin the motor control board might be defective. The motor control board provides power to the motor as well as direction and force. If the motor control board is defective the washer may not work at all or function properly.
If the washer won't spin the drive block might be worn out or may have come loose. Underneath the agitator the drive block connects the inner tub (the one that spins around) to the transmission. If the drive block wears out the transmission shaft will still spin but the tub may not - or it may spin slowly. This also usually causes a lot of noise. The drive block is relatively easy to replace.
If the washer won't spin, the clutch assembly might be worn out. The clutch assembly makes the connection between the transmission and the inner tub. It is designed to allow the tub to get up to the proper spin speed gradually. As the clutch wears out it may prevent the tub from spinning. The clutch is not repairable, if it is loud or not working properly it will need to be replaced.
If the washer won't spin the tub bearing might have seized. This is a common problem and relatively easy to replace.
If the washer won't spin the transmission might be worn out or defective. There are numerous more likely causes when a washer won't spin, which should be checked first.